Re: Programming is the Engineering Discipline of the Science that is Mathematics
Date: 9 Jun 2006 13:26:26 -0700
Keith H Duggar wrote:
> vc wrote:
> > Let P(A and B ) = P(A)*P(B) where P stands for
> > probabilities of respective events. Please 'reduce' the
> > above and tell when A and B is true, i.e. when P(A and B)
> > = 1.
> I assume by "and" you mean conjunction? Also do you realize
> that the "*" you wrote above is /not/ a connective? (which
> you claimed before it now seems).
You are confused. In any standard exposition of PT, the probability of two independent events is the product of the probabilities of those two events. In math, the product is designated by '*', not by 'and'.
>Here is the reduction:
> A : B : A and B
> 0 : 0 : 0
> 0 : 1 : 0
> 1 : 0 : 0
> 1 : 1 : 1
So your 'reduction' assigns the probability of one to both P(A) and P(B). Cute. So what kind of events are A and B if the probability of each of them is one ? Going in the opposite direction what would be the probability of 'A or B' ? Is it two ? If it's not two, what is it ?
> > > Third, +/* are not the connectives of PT. PT uses the
> > > same connectives as logic: conjunction, disjunction, and
> > > negation (whatever symbol you decide to give them).
> > Cool, so what are the truth tables for those connectives
> > in PT, or alternatively what are the derivation rules?
> For truth-valued statements they are exactly the same as
> those for logic! I wrote the one for conjunction above, I'm
> sure you know the others.
The probability statements are not truth-valued (true/false, 1/0), they have probability valuations. So what would be the probability valuation table for let's say implication ?
> > > Fourth, these connectives (same as logic remember) ARE truth
> > > functional in PT. That is when you apply the connectives to
> > > truth-valued statements you get truth-valued statements
> > > whose truth depends only on the constituent truth-values.
> > > (If you don't agree to this then provide a counter-example.)
> > Consider the real interval [0..1] with two subintervals A
> > and B whose respective lengths are say 1/3 and 1/8.
> > Assuming the uniform distribution, a randomly chosen point
> > would have the probability P(A) = 1/3 to be in subinterval
> > A and the probability P(B)=1/8 to be in subinterval B.
> > What is the probability P(A and B), the probability of the
> > randomly chosen point being both in A and B ?
> What part of "apply the connectives to truth-valued
> statements" did you miss? Furthermore, I have no idea what
> /statements/ A and B are supposed to be.
It's obvious that A is 'the randomly chosen point is in the subinterval A whose length is 1/3' and B is 'the randomly chosen point is in the subinterval B whose length is 1/8'.
> In PT just as in
> logic the connectives apply to /statements/ not lengths,
> points, subintervals, etc.
I asked you what is the probability of the conjunction (logical 'and') of two statements A and B, namely 'the randomly chosen point is in the subinterval A' and 'the randomly chosen point is in the subinterval B'.
> Regardless, you assigned
> probability rather than truth values. Try again with
> truth-valued statements.
I do not want to assign truth values because your original statement was: "Fourth, these connectives (same as logic remember) ARE truth functional in PT." Apparently, you've made a statement that the connectives are truth functional with respect to probability assignments. Did you you make your assertion with respect to probability assignments? yes or no ? If yes, do you still insist that P(A and B) depends just on P(A) and P(B) ? If you made your statement with respect to truth value assignments, you did not say anything relevant to PT.
> > > Just as when you apply the gamma function to natural
> > > numbers you get a natural numbers (no zero quibbles
> > > please).
> > >
> > > When you apply the connectives to a probability-valued
> > > statements you get probability-valued statements whose
> > > probability depends only on the constituent
> > > probabilities.
> > No.
> How informative.
So what is the probability of the event ('the randomly chosen point is in the subinterval A' and 'the randomly chosen point is in the subinterval B') assuming you know the the 'constituent' probabilities and relying on your assertion that "probability depends only on the constituent probabilities". ?
> > > Just as when you apply the gamma function to real
> > > numbers you get real numbers.
> > Forget the gamma, it's truly irrelevant.
> Once I know we agree on what "generalization" is.
> > > This is why PT is a /generalization/ of logic. It
> > > reduces to logic when applied to truth-valued
> > > statements.
> > So how about reducing the example I've given above ?
> I did, and it was warm and tart and trivial on my tongue.
You did not, see above.
> > > "Probability Theory: The Logic of Science" - ET Jaynes
> > That's unfortunate that Jaynes included a chapter on
> > similarities between logic and probabilistic reasoning in
> > his otherwise interesting book. If he did not, there
> > would have been much fewer confused readers.
> LOL. Amusing dismissal. I guess the title of his point is
> also confusing? Have you actually read the book or did you
> just look at a TOC after I mentioned it? Cause, if you have
> read it then you must realize that almost everything I have
> said here and the responses to your trivial challenges are
> all explained with much greater care in the that book and
> other sources. So can you point to flaws in his reasoning
> then? Since you are adamant that PT is not a generalization
> of logic perhaps you can point me to one of the surely
> numerous resources demonstrating this?
How about a) proving your statement that "probability depends only on the constituent probabilities" using my trival challenge; b) explaining what kind of events A and B are if P(A) = 1, P(B) = 1, and P(A and B) = 1 as well as what would be the probability of 'A or B' assuming your probability assignments of one.
When you've dealt successfully with (a) and (b), we can gradute to the book discussion. Received on Fri Jun 09 2006 - 22:26:26 CEST